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Phase space refers to the space in which dynamics occurs. In order to describe dynamics with a Hamiltonian, one must specify the positions and momenta, x→{\displaystyle {\vec {x}}} and p→{\displaystyle {\vec {p}}}. Although phase space in general may be a 2N dimensional manifold with non-trivial topology(a pendulum for example, has a position coordinate that connects back on itself). Usually, however, the phase space is R2N{\displaystyle \mathbb {R} ^{2N}}.

An observable or simply function is a function from the phase space to R{\displaystyle \mathbb {R} }. They can be represented as a multivariate polynomial or approximated by a truncated taylor series. An example of distribution is:

If a point in the face space has coordinates (x,p){\displaystyle (x,p)}, f(x,p){\displaystyle f(x,p)} is an observable, M=(m1(x,p),m2(x,p)){\displaystyle M=(m1(x,p),m2(x,p))} is a map where m1,m2{\displaystyle m1,m2} are observables, composition is defined by:

An operator is a function that transform a function in a function. A map is also an operator. Operators can be generated by function like derivative operators, vector fields, lie operator. Operator can be composed to form, for instance, exponential operators.

An algebra with the properties of the derivative. Related to field of non-standard analysis. TPSA vectors are approximate examples of. See also [1]

TPSA

Truncated power series algebra. Algebra of power series all truncated at a particular order. Power series may be added, multiplied. Analytic functions can be defined for them. A power series can be composed with a map. Example: epsilon(z).

k-Jets

Power series vector truncated at a particular order k{\displaystyle k}. A compositional map may be represented as a K-jets if the generating map maps the origin into the origin. See also [2]

compositional map

An operator generated by a map or a function equivalent to the composition of the map with another map. A compositional map may be represented as a k-jets if the generating map maps the origin into the origin.

Lie transformation

The transformation induced by a Lie operator by exponentiating. In particular, if :f: is a Lie operator, then e:f:{\displaystyle e^{:f:}} is a Lie transformation. The Lie Transformations form a group, a Lie group, which is also a topological group, when defined in a more general setting.

lie algebra

In general, any vector field that also has a multiplication property that satisfies

bilinear

anti-commutative

Jacobi identity

In classical dynamics, refers to either phase space functions with Poisson bracket as multiplication, or Lie operators with commutation as multiplication

Floquet space

Normalized space in which particles move in circles. Connected to Floquet's theorem which is more commonly known in solid state physics as Bloch's theorem. See also [3]

BCH formula

A formula relating the combining of two exponential operators into a single operator. For finite matrices, we state

eAeB=eC{\displaystyle e^{A}e^{B}=e^{C}}

where C is composed of sums of nested commutators of A and B. Due to the formula [:f:,:g:]=:{f,g}:, this generalizes in the case of Lie operators to the statement that

e:f:e:g:=e:h:{\displaystyle e^{:f:}e^{:g:}=e^{:h:}}

where h is a distribution on phase space. We note, however, that this is a purely formal relationship, and may in fact break down due to lack of convergence. h may be expressed in a series in different forms depending on what is considered the expansion parameter. If both f and g are considered small, then